Conchoid

The conchoid (the "shell-like" - like Latin concha from Greek κόγχη or κόγχος, shell ) is a special flat curve. It describes the movement of a point which - viewed from a fixed point ( pole ) - maintains a constant distance from a given curve.

Actual conchoid

It was already known in ancient Greece and is called the conchoid of Nicomedes after Nicomedes . Another name is Shell Curve . The name is derived from the fact that the graph resembles the two shells of a clam.

Three types of Nicomedes' conchoid

• Cartesian coordinates : ${\displaystyle (x-a)^{2}(x^{2}+y^{2})-b^{2}x^{2}\,=\,0}$

• Polar coordinates : ${\displaystyle r={\frac {a}{\cos \varphi }}\pm b}$

• Parameter representation : ${\displaystyle x(\varphi )=a+b\cos \varphi ;\qquad y(\varphi )=a\tan \varphi +b\sin \varphi }$

properties

• The points of the conchoid of Nicomedes are characterized by the following geometric property: Given are a straight line g ("leading straight line"), a point A that is at a distance of a from g (with a> 0), and a real number b (with b> 0). Then for an arbitrary point B of the straight line g the two points P and P ', which lie on the straight line AB and are at the distance b from B, lie on the conchoid.

Construction of the conchoid

• The falls are a type of curve that is known by the common name dog curve , especially for one branch that resembles the actual tractrix . ${\displaystyle b<=a}$ ${\displaystyle b=a}$

In the following it is assumed that the coordinate axes are as in the sketch, i.e. the pole is at the origin .

• The conchoid of the Nicomedes is axially symmetrical with respect to the x-axis. In general, three points of the curve lie on the axis of symmetry, namely , and the origin. ${\displaystyle (a+b|0)}$ ${\displaystyle (a-b|0)}$
  • For the origin is an isolated point . ${\displaystyle b<a}$
  • For two of the three points at the origin coincide, for the origin is a colon of the curve, so it is run through twice, the graph has a loop. ${\displaystyle b>=a}$ ${\displaystyle b>a}$
    • The two tangents at the origin have the equations

${\displaystyle y={\frac {\sqrt {b^{2}-a^{2}}}{a}}x}$and . ${\displaystyle y=-{\frac {\sqrt {b^{2}-a^{2}}}{a}}x}$

For both tangents coincide with the x-axis. So the origin is a real tip ${\displaystyle b=a}$

Ordinary conchoid

The term conchoid can be generalized:

A curve k ( leading curve ), a point A ( pole ) and a positive real number b are given. For any point B that lies on curve k, consider the two points that lie on straight line AB and are at distance b from B. The set of all these points is called the conchoid of the guide curve .

The simplest representation uses polar coordinates: If A is at the origin and if , then the equation of the ordinary conchoid is: ${\displaystyle r=f(\varphi )}$
${\displaystyle r=f(\varphi )\pm b}$

properties

All common conchoids are cissoids , with one curve being a circle at the origin.

A Pascal snail is a conchoid, where the given curve is a circle.

General conchoid

If one extends the formation rule by not plotting the distance b along the straight line AB, but along a straight line that has a constant angle to AB at point B , one obtains the general conchoid . In the case of and the usual conchoid results, otherwise one speaks of a crooked conchoid . ${\displaystyle \alpha }$ ${\displaystyle \alpha =0}$ ${\displaystyle \alpha =\pi }$

Conchoidal toothing in gear technology

In the transmission art is the so called Konchoidenverzahnung one of several techniques for teeth of gears and toothed racks .

Remarks

• De Sluze's so-called conchoid is actually a special cissoid .
• Dürer's so-called conchoid is a more general construction.

Web links

Commons : Conchoid  - collection of images, videos and audio files

• Johanneum Lüneburg dog curve or conchoid of Nicomedes ( Memento from July 3, 2007 in the Internet Archive )